Reducing Curse of Dimensionality: Improved PTAS for TSP (with Neighborhoods) in Doubling Metrics

We consider the Traveling Salesman Problem with Neighborhoods (TSPN) in doubling metrics. The goal is to find a shortest tour that visits each of a given collection of subsets (regions or neighborhoods) in the underlying metric space. We give a randomized polynomial time approximation scheme (PTAS) when the regions are fat weakly disjoint. This notion of regions was first defined when a QPTAS was given for the problem in [SODA 2010: Chan and Elbassioni]. We combine the techniques in the previous work, together with the recent PTAS for TSP [STOC 2012: Bartal, Gottlieb and Krauthgamer] to achieve a PTAS for TSPN. Moreover, more refined procedures are used to improve the dependence of the running time on the doubling dimension k from the previous exp[O(1)k2 ] (even for just TSP) to exp[O(1)O(k log k) ].

• Type: Journal
• Year: 2016
• Where: SODA
• Authors: T.-H. Hubert Chan, Shaofeng H.-C. Jiang
• Keywords: Algorithms, SODA 2016